Limit of a sequence

In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the $\lim$ symbol (e.g., $\lim _{n\to \infty }a_{n}$ ).^[1] If such a limit exists, the sequence is called convergent.^[2] A sequence that does not converge is said to be divergent.^[3] The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.^[1]

For the general mathematical concept, see Limit (mathematics).

Limits can be defined in any metric or topological space, but are usually first encountered in the real numbers.

History[edit]

The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes.

Leucippus, Democritus, Antiphon, Eudoxus, and Archimedes developed the method of exhaustion, which uses an infinite sequence of approximations to determine an area or a volume. Archimedes succeeded in summing what is now called a geometric series.

Grégoire de Saint-Vincent gave the first definition of limit (terminus) of a geometric series in his work Opus Geometricum (1647): "The terminus of a progression is the end of the series, which none progression can reach, even not if she is continued in infinity, but which she can approach nearer than a given segment."^[4]

Pietro Mengoli anticipated the modern idea of limit of a sequence with his study of quasi-proportions in Geometriae speciosae elementa (1659). He used the term quasi-infinite for unbounded and quasi-null for vanishing.

Newton dealt with series in his works on Analysis with infinite series (written in 1669, circulated in manuscript, published in 1711), Method of fluxions and infinite series (written in 1671, published in English translation in 1736, Latin original published much later) and Tractatus de Quadratura Curvarum (written in 1693, published in 1704 as an Appendix to his Optiks). In the latter work, Newton considers the binomial expansion of ${\textstyle (x+o)^{n}}$ , which he then linearizes by taking the limit as ${\textstyle o}$ tends to ${\textstyle 0}$ .

In the 18th century, mathematicians such as Euler succeeded in summing some divergent series by stopping at the right moment; they did not much care whether a limit existed, as long as it could be calculated. At the end of the century, Lagrange in his Théorie des fonctions analytiques (1797) opined that the lack of rigour precluded further development in calculus. Gauss in his etude of hypergeometric series (1813) for the first time rigorously investigated the conditions under which a series converged to a limit.

The modern definition of a limit (for any ${\textstyle \varepsilon }$ there exists an index ${\textstyle N}$ so that ...) was given by Bernard Bolzano (Der binomische Lehrsatz, Prague 1816, which was little noticed at the time), and by Karl Weierstrass in the 1870s.

If $x_{n}=c$ for constant ${\textstyle c}$ , then $x_{n}\to c$ .^[5]

[proof 1]

If $x_{n}={\frac {1}{n}}$ , then $x_{n}\to 0$ .^[5]

[proof 2]

If $x_{n}={\frac {1}{n}}$ when $n$ is even, and $x_{n}={\frac {1}{n^{2}}}$ when $n$ is odd, then $x_{n}\to 0$ . (The fact that $x_{n+1}>x_{n}$ whenever $n$ is odd is irrelevant.)

Given any real number, one may easily construct a sequence that converges to that number by taking decimal approximations. For example, the sequence ${\textstyle 0.3,0.33,0.333,0.3333,\dots }$ converges to ${\textstyle {\frac {1}{3}}}$ . The ${\textstyle 0.3333\dots }$ is the limit of the previous sequence, defined by $0.3333...:=\lim _{n\to \infty }\sum _{k=1}^{n}{\frac {3}{10^{k}}}$

decimal representation

Finding the limit of a sequence is not always obvious. Two examples are $\lim _{n\to \infty }\left(1+{\tfrac {1}{n}}\right)^{n}$ (the limit of which is the ) and the arithmetic–geometric mean. The squeeze theorem is often useful in the establishment of such limits.

number e

Metric spaces[edit]

Definition[edit]

A point $x$ of the metric space $(X,d)$ is the limit of the sequence $(x_{n})$ if:

Topological spaces[edit]

Definition[edit]

A point $x\in X$ of the topological space $(X,\tau )$ is a limit or limit point^[7]^[8] of the sequence $\left(x_{n}\right)_{n\in \mathbb {N} }$ if:

If $x_{n,m}=c$ for constant ${\textstyle c}$ , then $x_{n,m}\to c$ .

If $x_{n,m}={\frac {1}{n+m}}$ , then $x_{n,m}\to 0$ .

If $x_{n,m}={\frac {n}{n+m}}$ , then the limit does not exist. Depending on the relative "growing speed" of ${\textstyle n}$ and ${\textstyle m}$ , this sequence can get closer to any value between ${\textstyle 0}$ and ${\textstyle 1}$ .

Limit point

Subsequential limit

Limit superior and limit inferior

Limit of a function

Limit of a sequence of functions

Limit of a sequence of sets

Limit of a net

Pointwise convergence

Uniform convergence

Modes of convergence

(1978). General topology. Translated by Császár, Klára. Bristol England: Adam Hilger Ltd. ISBN 0-85274-275-4. OCLC 4146011.

Császár, Ákos

(1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485.

Dugundji, James

(1961). "Differential and Integral Calculus Volume I", Blackie & Son, Ltd., Glasgow.

Courant, Richard

and James Harkness A treatise on the theory of functions (New York: Macmillan, 1893)

Frank Morley

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

Limit of a sequence

History[edit]

[proof 1]

[proof 2]

decimal representation

number e

Metric spaces[edit]

Definition[edit]

Topological spaces[edit]

Definition[edit]

Limit point

Subsequential limit

Limit superior and limit inferior

Limit of a function

Limit of a sequence of sets

Limit of a net

Pointwise convergence

Uniform convergence

Modes of convergence

Császár, Ákos

Dugundji, James

Courant, Richard

Frank Morley

"Limit"

A history of the calculus, including limits